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For a square matrix A, if

AB = BA = I


Then, B is the inverse of A

i.e. B = A −1


We will find inverse of a matrix by

Note: Since AB = BA = I

We can say B is the inverse of A.

i.e. B = A −1


We can also say,

A is the inverse of B

i.e. A = B −1

Thus, for inverse

We can write

  AA −1 = A −1 A = I

Where I is identity matrix of the same order as A


Let’s look at same properties of Inverse.


Properties of Inverse

  1. For  a matrix A,
    A −1 is unique, i.e., there is only one inverse of a matrix
  2. (A −1 ) −1   = A
  3. (kA) -1 = 1/k A -1  

    Note: This is different from

    (kA) T = k A T

  4. (A -1 ) T = (A T ) -1

  5. (A + B) -1 = A -1 + B -1

  6.  (AB) -1 = B -1 A -1



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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.